A220881 Number of nonequivalent dissections of an n-gon into n-3 polygons by nonintersecting diagonals up to rotation.

1, 1, 4, 12, 43, 143, 504, 1768, 6310, 22610, 81752, 297160, 1086601, 3991995, 14732720, 54587280, 202997670, 757398510, 2834510744, 10637507400, 40023636310, 150946230006, 570534578704, 2160865067312, 8199711378716, 31170212479588, 118686578956272

Offset: 4

Author

N. J. A. Sloane, Dec 28 2012

Keywords

• nonn

Comments

This is almost identical to A003444, but has a different offset and a more precise definition.

In other words, the number of almost-triangulations of an n-gon modulo the cyclic action.

Equivalently, the number of edges of the (n-3)-dimensional associahedron modulo the cyclic action.

The dissection will always be composed of one quadrilateral and n-4 triangles. - Andrew Howroyd, Nov 25 2017

Also number of necklaces of 2 colors with 2n-4 beads and n black ones. - Wouter Meeussen, Aug 03 2002

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• D. Bowman and A. Regev, Counting symmetry classes of dissections of a convex regular polygon, arXiv:1209.6270 [math.CO], 2012.
• P. Lisonek, Closed forms for the number of polygon dissections, Journal of Symbolic Computation 20 (1995), 595-601.
• Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978) 370-388.

Crossrefs

A diagonal of A295633.

Cf. A003444, A003445.

Cf. A003442, A003447, A003449.

Programs

Maple

C:=n->binomial(2*n,n)/(n+1);
T2:= proc(n) local t1; global C;
t1 :=  (n-3)*C(n-2)/(2*n);
if n mod 4 = 0 then t1:=t1+C(n/4-1)/2 fi;
if n mod 2 = 0 then t1:=t1+C(n/2-1)/4 fi;
t1; end;
[seq(T2(n),n=4..40)];

Mathematica

c[n_] := Binomial[2*n, n]/(n+1);
T2[n_] := Module[{t1}, t1 = (n-3)*c[n-2]/(2*n); If[Mod[n, 4] == 0, t1 = t1 + c[n/4-1]/2]; If[Mod[n, 2] == 0, t1 = t1 + c[n/2-1]/4]; t1];
Table[T2[n], {n, 4, 40}] (* Jean-François Alcover, Nov 23 2017, translated from Maple *)
a[n_] := Sum[EulerPhi[d]*Binomial[(2n-4)/d, n/d], {d, Divisors[GCD[2n-4, n] ]}]/(2n-4);
Array[a, 30, 4] (* Jean-François Alcover, Dec 02 2017, after Andrew Howroyd *)

PARI

a(n) = if(n>=4, sumdiv(gcd(2*n-4, n), d, eulerphi(d)*binomial((2*n-4)/d, n/d))/(2*n-4)) \\ Andrew Howroyd, Nov 25 2017

Formula

a(n) = (1/(2n-4)) Sum_{d |(2n-4, n)} phi(d)*binomial((2n-4)/d, n/d) for n >= 4. - Wouter Meeussen, Aug 03 2002

Extensions

Name clarified by Andrew Howroyd, Nov 25 2017